Universität Koblenz-Landau FB 4 Informatik
Prof. Dr. Viorica Sofronie-Stokkermans∗1 05.07.2021 M.Ed. Dennis Peuter∗2
Exercises for Advances in Theoretical Computer Science Exercise Sheet 12
Due at 13.07.2021, 10:00 s.t.
Exercise 12.1
Prove that for every alphabetΣwith|Σ| ≥2it is undecidable whether for DCFL languages L1, L2 we have L1 ⊆L2.
Hint: Reduction to the problem of testing emptiness for intersection of DFCL languages.
Use the fact that the complement of a DCFL language is a DCFL language.
Exercise 12.2
Definition: A map h : Σ∗1 → Σ∗2 is a monoid homomorphism if it has the property that for all words w1, w2 ∈Σ∗1, h(w1w2) =h(w1)h(w2).
Prove that the following problem is undecidable:
Letf, g: Σ∗1 →Σ∗2 be monoid homomorphisms. Assume that Σ1 ={a1, . . . , an}. Is there a wordw∈Σ∗1 such thatf(w) =g(w)?
Hint: Note that f andg are completely described by their values on a1, . . . , an.
Exercise 12.3
Definitions: Assume we are in propositional logic with propositional variablesΠ.
• A literal L is a propositional variable P or the negation of a propositional variable
¬P.
• A propositional formula is in disjunctive normal form (DNF) if it has the form (L11∧ · · · ∧L1n1)∨ · · · ∨(Lm1 ∧ · · · ∧Lmnm).
• A propositional formula is aclauseif it is of the formL1∨· · ·∨Ln(i.e. is a disjunction of literals). AHorn clauseis a clause which contains at most one positive literal. (For instance P∨¬Q∨¬R and ¬Q∨¬R are Horn clauses, but P∨Q∨¬R is not.)
a) Is it true that for every formulaF in disjunctive normal form we can check whetherF is satisfiable in polynomial time? Briefly justify your answer.
b) Is it true that for every formula F which is a conjunction of Horn clauses we can check whetherF is satisfiable in polynomial time? Briefly justify your answer.
Remark:For answering these questions you do not need to construct Turing machines. You can use results on propositional logic presented e.g. in the lecture “Logik für Informatiker”
(e.g. the existence of algorithms for checking the satisfiability of sets (= conjunctions) of Horn clauses).
∗1
B 225 sofronie@uni-koblenz.de https://userpages.uni-koblenz.de/~sofronie/
∗2
B 223 dpeuter@uni-koblenz.de https://userpages.uni-koblenz.de/~dpeuter/
If you want to submit solutions, please do so until 13.07.2021, 10:00 s.t. via e-mail (with “Homework ACTCS” in the subject) todpeuter@uni-koblenz.de.
